A class of exactly solvable models for the Schrödinger equation

• Authors: Charles Downing
• Languages of publication: EN

Abstracts

EN

We present a class of confining potentials which allow one to reduce the one-dimensional Schrödinger equation to a named equation of mathematical physics, namely either Bessel’s or Whittaker’s differential equation. In all cases, we provide closed form expressions for both the symmetric and antisymmetric wavefunction solutions, each along with an associated transcendental equation for allowed eigenvalues. The class of potentials considered contains an example of both cusp-like single wells and a double-well.

Keywords

EN

solutions of wave equations: bound states; Schrodinger equation

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 8
• Pages: 977-983

Dates

published

1 - 8 - 2013

online

23 - 10 - 2013

Contributors

author

Charles Downing

• School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK

References

• [1] E. Schrödinger, Ann. Phys. (Leipzig) 79, 361 (1926) http://dx.doi.org/10.1002/andp.19263840404[Crossref]
• [2] P. M. Morse, Phys. Rev. 34, 57 (1929) http://dx.doi.org/10.1103/PhysRev.34.57[Crossref]
• [3] C. Eckart, Phys. Rev. 35, 1303 (1930) http://dx.doi.org/10.1103/PhysRev.35.1303[Crossref]
• [4] N. Rosen, P. M. Morse, Phys. Rev. 42, 210 (1932) http://dx.doi.org/10.1103/PhysRev.42.210[Crossref]
• [5] G. Pöschl, E. Teller, Z. Phys. 83, 143 (1933) http://dx.doi.org/10.1007/BF01331132[Crossref]
• [6] A list of exact solutions from the genesis of quantum mechanics is given in M. F. Manning, Phys. Rev. 48, 161 (1935) http://dx.doi.org/10.1103/PhysRev.48.161[Crossref]
• [7] F. Scarf, Phys. Rev. 112, 1137 (1958) http://dx.doi.org/10.1103/PhysRev.112.1137[Crossref]
• [8] R. Loudon, Am. J. Phys. 27, 649 (1959) http://dx.doi.org/10.1119/1.1934950[Crossref]
• [9] R. R. Whitehead, A. Watt, G. P. Flessas, M. A. Nagarajan, J. Phys. A: Math. Gen. 15, 1217 (1982) http://dx.doi.org/10.1088/0305-4470/15/4/024[Crossref]
• [10] D. Pertsch J. Phys. A: Math. Gen. 23, 4145 (1990) http://dx.doi.org/10.1088/0305-4470/23/18/020[Crossref]
• [11] C. M. Bender, Q. Wang, J. Phys. A: Math. Gen. 34 9835 (2001) http://dx.doi.org/10.1088/0305-4470/34/46/307[Crossref]
• [12] D. G. W. Parfitt, M. E. Portnoi, J. Math. Phys. 43, 4681 (2002) http://dx.doi.org/10.1063/1.1503868[Crossref]
• [13] For a comprehensive list of exact solutions please see V. G. Bagrov, D. M. Gitman, Exact Solutions of Relativistic Wave Equations (Kluwer, Dordrecht, 1990) http://dx.doi.org/10.1007/978-94-009-1854-2[Crossref]
• [14] L. Infeld, T. E. Hull, Rev. Mod. Phys. 23, 21 (1951) http://dx.doi.org/10.1103/RevModPhys.23.21[Crossref]
• [15] F. Cooper, A. Khare, U. P. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singapore, 2001) http://dx.doi.org/10.1142/4687[Crossref]
• [16] A. Gangopadhyaya, J. V. Mallow and C. Rasinariu, Supersymmetric Quantum Mechanics (World Scientific, Singapore, 2011), and references therein
• [17] A. V. Turbiner, Sov. Phys. JETP 67, 230 (1988)
• [18] A. Turbiner, Commun. Math. Phys. 118, 467 (1988) http://dx.doi.org/10.1007/BF01466727[Crossref]
• [19] A. G. Ushveridze, Quasi-exactly Solvable Models in Quantum Mechanics (Institute of Physics, Bristol, 1994)
• [20] C. M. Bender, S. Boettcher, J. Phys. A 31, L273 (1998) http://dx.doi.org/10.1088/0305-4470/31/14/001[Crossref]
• [21] C. A. Downing, J. Math. Phys. 54, 072101 (2013) http://dx.doi.org/10.1063/1.4811855[Crossref]
• [22] R. R. Hartmann, arXiv:1306.2836 (2013) [WoS]
• [23] M. J. Gillan, J. Phys. C: Solid State Phys. 20 3621 (1987) http://dx.doi.org/10.1088/0022-3719/20/24/005[Crossref]
• [24] Z. I. Alferov, Rev. Mod. Phys. 73, 767 (2001) http://dx.doi.org/10.1103/RevModPhys.73.767[Crossref]
• [25] P. Budau, M. Grigorescu, Phys. Rev. B 57, 6313 (1998) http://dx.doi.org/10.1103/PhysRevB.57.6313[Crossref]
• [26] R. W. Spekkens, J. E. Sipe, Phys. Rev. A 59, 3868 (1999) http://dx.doi.org/10.1103/PhysRevA.59.3868[Crossref]
• [27] S. Coleman, Aspects of Symmetry (Cambridge University Press, Cambridge, 1985) http://dx.doi.org/10.1017/CBO9780511565045[Crossref]
• [28] K. Heun Math. Ann. 33, 161 (1889) http://dx.doi.org/10.1007/BF01443849[Crossref]
• [29] A. Ronveaux, Heun’s Differential Equations (Oxford University Press, Oxford, 1995)
• [30] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972)
• [31] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980)

Identifiers

DOI

10.2478/s11534-013-0301-6